One to one, onto, and bijective functions make sense. For me, the waters get murky when we throw in the Real Numbers. Because the real numbers are infinite sets, it gets confusing to me how a function can be one-to-one but not onto or vice versa. It would seem like the domain and codomain in each instance are infinite and so by the same reasons that a function is not onto it would also not be one to one.
Something about proving that something is bijective is just satisfying. It's like, no matter which way I go from domain to codomain I know what's going on. Each element in the domain is unique and the same for codomain. That brings up a practical application question though, would this ever truly happen in a set of data? I suppose if you were measuring the velocity of some constantly increasing object you would have distinct time and velocity data. Is there anything cool that we can use bijective functions for?
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